The Math Behind Dante's Divine Comedy

 Number and logic underlie the entire Commedia. All of Dante’s universe comprises a strikingly robust, sophisticated physical structure, reflecting a deeper moral and theological structure, all communicated through an intricate poetic structure. It is a highly measured project, the specifics of which can be explored and explained seemingly to no end. Even in the introduction to his Vita nuova, Dante understands and describes his first encounter with Beatrice as an event of cosmological significance: “Dante had begun his evocation of Beatrice with resplendent astronomical allusions.”[1] The motivation and the goal for his Commedia are deeply rooted in a logical and measured view of the cosmos.

These specifics are certainly important in Dante’s universe, as they reflect and communicate truths about the logical order inherent in Creation. Dante encounters these structures and the space through which he moves in a very intimate, sensory way. In the Paradiso, as Dante journeys toward his vision of the Divine, the focus on number and structure does not dwindle, but the importance shifts from understanding the components of these intricate structures of the universe to instead appreciating the paradoxically simple source and ground of being that is God. Dante communicates the conception of the Divine as a point, a punto, through number, geometry (number applied to space), physics (number applied to space and time), and optics (physics applied to sight). This culminates in the Primo Mobile, where Dante is overcome by the dimensionless punto that is the source and ground of being.

One can understand Dante’s journey both as originating from the center of the material universe and emanating outwards, and also as a journey inward to a dimensionless point giving rise to all of reality. This is a paradox: Dante moves outward, always to the next realm of the universe, while also converging on a single point. The poem, of course, relates his trek through Hell, Purgatory, and finally Paradise, and renders each as physical locations at least insofar as Dante moves through them. As Dante (and the reader) grows spiritually in his journey, though, the journey becomes more apparently a journey to the Divine in a single point. As he proceeds, Dante’s vision improves and becomes more ready to grasp the paradox of his journey. Jennifer Martin describes this evolution of vision in the Paradiso: “Dante’s Paradiso is, at its most fundamental, a text about this kind of wounded seeing that is led toward truer and truer sight, where the subjective and objective converge in an experience of God that cannot be sufficiently pronounced.”[2] The seer, what is seen, and the very act of seeing must all coincide.

Early in the Paradiso, Dante confronts the problem of multiplicity and differentiation in Creation. He first poses the question of differentiation in Par. II. 49-51, when he asks Beatrice about the dark spots on the moon: “Ma ditemi: che son li segni bui / di questo corpo, che là giuso in terra / fan di Cain favoleggiare altrui?” [But now tell me: / what are the dark marks on this planet’s body / that there below, on earth, have made men tell / the tale of Cain?”] What Dante notices is difference, that is, the lack of uniformity. He believes the spots to be due to alternating density and rarity of matter. Beatrice, to show Dante his error, now proposes an experiment with three mirrors: two mirrors are placed side by side with a third between the two and farther back. A candle is placed behind the performer of the experiment, who can observe that the reflected brightness is constant in each mirror. The middle reflection is smaller but not less bright.

The mirror experiment is supremely important for a few reasons. For one, as Allen Tate suggests, the size discrepancy of the reflections denotes unequal reception of a quality, while the equal brightness denotes equality of salvation.[3] Dante may perceive difference, but underlying the multiplicity is a deeper unity. Piccarda, the soul Dante encounters in the following canto, reiterates this truth. When Dante inquires if Piccarda would rather be in a higher circle of Paradise, she assures him that each soul desires no more, that this is the essence of salvation: that all wills coincide with God’s.

Beyond this spiritual truth communicated by the result of the experiment, Mark Peterson observes that “it is the single largest intrusion of physical theory into the poem.”[4] He also notes that Dante’s physical discourse here is, in fact, correct and original: “The invariance of apparent brightness with distance is, so far as I know, original with Dante, an original discovery, and not at all obvious.”[5] Thus, early in the Paradiso, Dante makes clear the importance he places on using symbolism that is revealing of ultimate reality while also remaining grounded in material reality.

Furthermore, the mirror experiment introduces the key themes of light, reflection, optics, and geometry (to name a few), very early in the Paradiso. Although Dante is “more sparing in his use of the mirror metaphor in the final cantica,”[6] it is a useful image for Dante and the reader to keep in mind as Dante journeys to God. As “Dante-personaggio travels through the light-filled celestial spheres towards God as source of light . . . everything is potentially a light-reflector, the blessed souls, planets and stars, angels, Beatrice, and even the convex surface of the ninth heaven, the Primum Mobile.”[7]

The experiment’s setup even suggests Trinitarian imagery, both visually and dramatically. Visually, the middle mirror, the performer of the experiment, and the candle all lie on a vertical axis; the two side mirrors lie on a horizontal axis. The two axes intersect in the form of a cross. Dramatically, Allen Tate points out that “there are three mirrors reflecting the one light,”[8] an obvious nod to the Trinity. Both of these previous points are even more convincing when one realizes that the experiment would have demonstrated the same result with only two mirrors placed at different distances from the candle. The mirror experiment is thus a convergence of mathematical, geometrical, optical, and spiritual principles. It is a seed that Dante—and the reader—must keep in mind on the journey to the Divine.

On the journey through Paradise, “everything . . . is a reflecting light and it is this light which Dante uses to represent substance, which light is not a passive reflection of an external source but an active reflection of internal vision.”[9] Thus, everything on Dante’s journey becomes something that reflects. James Miller holds that, keeping the mirror metaphor in mind, “Creation becomes a hierarchy of mirrors, each casting an image of God. The farther off the individual image is from God, the smaller the image of Him it reflects, but his brightness never diminishes.”[10]

Reflection must be treated while simultaneously considering convergence. When a human being observes himself in a mirror, his image is returning to its source, to its ground, and is being perceived by his own eye. One can imagine this observer moving closer and closer to the mirror until he is touching it. At such a close distance from the mirror, the image now travels very little to return to its observer. The source and the reflection have nearly fully converged.

As Dante travels through Paradise, he undergoes a process of self-reflection and of seeing reflections in other things. The process is similar to what was described for the observer moving closer to the mirror, with a fundamental difference: the source and the reflection finally do fully converge as Dante moves toward the Empyrean. That is, all becomes encompassed within the single, dimensionless point that gives rise to all of Creation. There is no more reflection in the sense that it is construed materially, as this implies a separation between source and reflector. The tools to begin this mental exercise were all present in the mirror experiment. In the Primo Mobile, this idea explodes. Tate reflects,

What Dante sees in the Primum Mobile is this perspective[11] visually reversed; instead of being the outer “crust” of the universe, the Primum Mobile is actually next to the central Still Point, whirling with inconceivable speed. God, the Still Point, is a non-spatial entity which is everywhere and nowhere.[12]

In Par. XXVII. 100-102, Dante notes that the Primo Mobile has no distinguishable features, just speed: “Le parti sue vivissime ed eccelse / si uniforme son, ch’i’ non so dire / qual Bëatrice per loco mi scelse ” [Its parts were all so equally alive / and excellent, that I cannot say which / place Beatrice selected for my entry]. Again, Dante returns to the subject of the physical world (speed) to describe the Primo Mobile, where consciousness originates. Speed—rendered mathematically as space (distance) against time—arises from consciousness, from the Primo Mobile. But Beatrice explains, in lines 109-110, “e questo cielo non ha altro dove / che la mente divina” [This heaven has no other where than this: / the mind of God]. The Divine mind, consciousness–which, as Tate argued, is both everywhere and nowhere—gives rise to all of space-time.

Beatrice continues, just a few lines later in Par. XXVII. 115-117, “Non è suo moto per altro distinto, / ma li altri son mensurati da questo, / sì come diece da mezzo e da quinto” [No other heaven measures this sphere’s motion, / but it serves as the measure for the rest / even as half and fifth determine ten]. Directly after an explication of the Primo Mobile from physics, Beatrice brings up prime factorization, to the same end: all motion originates in the Primo Mobile and is measured against the motion of the Primo Mobile in the same way that ten arises from its prime factors, five and two.

The idea forwarded by Beatrice boils down to a deeply mathematical idea: that numbers can be assembled by their factors; that the number one gives rise to all numbers; indeed, that the number one by itself stands for the entire concept of number and then of mathematics.[13] Thus, that ten arises from two and five adds nothing to what is already encompassed in the number one. In the same way, the world, material reality, adds nothing to the all-encompassing reality of the Empyrean.

This said, the numbers that Beatrice uses here have a deeper meaning. Christian Moevs explains the significance of two, five, and ten: two is the number of change and movement, five is the number of the world, and ten is the perfect number, the number of the Empyrean.[14] In her factorization,

Perhaps Beatrice is suggesting that to factor ten, the atemporal, aspatial stillness and fullness of the Empyrean, can be factored into movement, that is, into the duality or difference of an original state and an altered state, which is already to have the other factor, five: the duality of mover and moved, the duality of before and after, and one with these, time.[15]

Beatrice’s explanation of the Primo Mobile and the Empyrean is at once physical, mathematical, and metaphysical, all in concert to describe the singularity that is the Divine, which is everywhere and nowhere, which is beyond any conceived notion of space or time but is without dimension.

In Canto 28, Dante sees a point of light reflected in Beatrice’s eyes, and he turns to gaze upon the source with his own eyes. Moevs notes, “By turning to see directly what (or as) Beatrice sees, Dante is assimilating his sight, or point of view, to hers. In absolute terms, to turn from the reflection to the source is to turn from the world to its ground; it is to focus the light of awareness on itself in a single point.”[16] When Dante observes what caused him to turn, Beatrice offers him an explanation of what he sees. Peter Dronke summarizes what happens here: “In Canto 28, at the center of the nine concentric circles of angels, is the divine punto, infinitely small and infinitely radiant–the point from which, as Beatrice (tacitly citing Aristotle) explains, the heavens and the whole of nature depend.”[17] The punto as characterization of the Divine has here become explicit.

In Paradiso 29, the punto appears again, this time recalling Francesca from Inferno 5. In what is a perfectly balanced pivot structure comprising the first 12 lines, Dante writes (lines 8-9), “si tacque Bëatrice, riguardando / fiso nel punto che m’avëa vinto” [Beatrice, a smile upon / her face, keep silent, even as she gazed / intently at the Point that overwhelmed me]. In Inferno 5, Francesca had been overcome by a single point in her poor reading of Lancelot, and at this point she was drawn into sin. Dante, too, is overcome by a point here. The point that overcomes Dante, in contrast, is the Divine. It is interesting to note a similar dichotomy that exists with Satan and the Empyrean: “Only Satan, at the geometrical center of the world, occupies a point that cannot be located on any existing arc of the cosmos.”[18] Satan occupies a finite, ultimately nonexistent point in the center of the world. In contrast, the Divine does not “occupy” any point; it is the dimensionless point outside of space and time that gives rise to infinity. Accordingly, the point that overcomes Francesca in Inferno 5 is ultimately illusory, only drawing her deeper into her own finitude. The point that here overcomes Dante reveals to him the infinite possibility and the infinite goodness of the Divine.

The reader is sent back again and again to the beautifully simple, paradoxical, even mathematical concept of the point: paradoxical because the point is, by definition, dimensionless, and yet it gives rise to the infinite. I have always marveled at a foundational concept in introductory calculus: an infinitesimally small unit of area or volume is used to find a finite area or volume; the infinite is contained within the borders of the finite. Dante illustrates in words that which is difficult to conceive even visually: his journey through the Paradiso is a constant reflection, an ever-shrinking loop of turning back the mind on Creation, on himself, until the loop shrinks down to this dimensionless point, from which is born infinity.

As God’s Creation, we experience a physical realm of differentiated entities and perceive multiplicity in our material reality. The character of Beatrice utilizes this fact in Paradiso 2 when she proposes the mirror experiment. The experiment combines mathematical, geometrical, and optical/physical principles to demonstrate spiritual truths. This experiment, especially its utilization of reflection, plants a seed in Dante, prodding him on his journey to the Divine: “Nature offers to the symbolic poet clearly denotable objects in depth and in the round, which yield the analogies to the higher senses.”[19] In the Primo Mobile, Dante the poet utilizes these same principles as he approaches the dimensionless punto of the Divine, the source and ground of all being.

[1] Dronke, “Symbolism and Structure in Paradiso 30,” 33.

[2] Martin, “Geographics of Stars, Metaphysics of Light,” 142.

[3] Tate, “The Symbolic Imagination,” 276.

[4] Peterson, “Dante’s Physics,” 170.

[5] Ibid., 171.

[6] Gilson, “Light Reflection, Mirror Metaphors, and Optical Framing in Dante’s Comedy,” 241.

[7] Ibid., 241.

[8] Tate, “The Symbolic Imagination,” 276.

[9] Chiarenza, “The Imageless Vision and Dante’s Paradiso,” 83.

[10] Miller, “Three Mirrors of Dante’s Paradiso,” 266.

[11] That is, the perspective of the concentric circles that form Paradise.

[12] Tate, “The Symbolic Imagination,” 271.

[13] Perhaps too mathematical for the scope of this paper, but in group theory, this idea is rather fundamental: we say that 1 is a “generator” for the infinite cyclic group of integers. Basically, all that is needed to generate the entire group of integers (…, -3, -2, -1, 0, 1, 2, 3, …) is 1.

[14] Moevs, The Metaphysics of Dante’s Comedy, 138-139.

[15] Ibid., 139.

[16] Ibid., 141.

[17] Dronke, “Symbolism and Structure in Paradiso 30,” 31.

[18] Tate, “The Symbolic Imagination,” 270.

[19] Tate, “The Symbolic Imagination,” 262.

Featured Image: Giovanni Guida, Apotheosis of Dante Alighieri in Florence, 2021 ; Source: Wikimedia Commons, CC BY-SA 3.0.


Matthew Canonico

Matthew Canonico is a 2020 graduate of the University of Notre Dame, where he studied mathematics and Italian.

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